25 research outputs found
A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations
In this paper a hybridized weak Galerkin (HWG) finite element method for
solving the Stokes equations in the primary velocity-pressure formulation is
introduced. The WG method uses weak functions and their weak derivatives which
are defined as distributions. Weak functions and weak derivatives can be
approximated by piecewise polynomials with various degrees. Different
combination of polynomial spaces leads to different WG finite element methods,
which makes WG methods highly flexible and efficient in practical computation.
A Lagrange multiplier is introduced to provide a numerical approximation for
certain derivatives of the exact solution. With this new feature, HWG method
can be used to deal with jumps of the functions and their flux easily. Optimal
order error estimate are established for the corresponding HWG finite element
approximations for both {\color{black}primal variables} and the Lagrange
multiplier. A Schur complement formulation of the HWG method is derived for
implementation purpose. The validity of the theoretical results is demonstrated
in numerical tests.Comment: 19 pages, 4 tables,it has been accepted for publication in SCIENCE
CHINA Mathematics. arXiv admin note: substantial text overlap with
arXiv:1402.1157, arXiv:1302.2707 by other author
The stabilizer free weak Galerkin mixed finite elements method for the biharmonic equation
In this article, the stabilizer free weak Galerkin (SFWG) finite element
method is applied to the Ciarlet-Raviart mixed form of the Biharmonic equation.
We utilize the SFWG solutions of the second elliptic problems to define
projection operators, build error equations, and further derive the error
estimates. Finally, numerical examples support the results reached by the
theory
Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D
We consider the singularly perturbed fourth-order boundary value problem
on the unit square , with boundary conditions on
, where is a small parameter. The
problem is solved numerically by means of a weak Galerkin(WG) finite element
method, which is highly robust and flexible in the element construction by
using discontinuous piecewise polynomials on finite element partitions
consisting of polygons of arbitrary shape. The resulting WG finite element
formulation is symmetric, positive definite, and parameter-free. Under
reasonable assumptions on the structure of the boundary layers that appear in
the solution, a family of suitable Shishkin meshes with elements is
constructed ,convergence of the method is proved in a discrete norm for
the corresponding WG finite element solutions and numerical results are
presented